Avkhadiev–Lehto type constants in the study of the Gakhov class

Avkhadiev’s classes (of holomorphic functions with two-sided bounds of the modulus of the derivative) are studied in domains other than a unit disk. We give the conditions that ensure the uniqueness of the critical point of the conformal radius for the images of the mentioned domains under the mappings by the functions of the Avkhadiev classes. We use an analogue of the setting proposed at the time by O. Lehto to study the univalence of functions satisfying the conditions of the Nehari type in domains conformally equivalent to a disk.